Question 1129742
breakdown to the exact value of the roots:

{{{x^5-4x^4-7x^3+14x^2-44x+120 =0}}}

{{{x^5-2x^4-2x^4-11x^3+4x^3+22x^2-8x^2-60x+16x+120=0}}}

{{{(x^5-2x^4)-(11x^3-22x^2)-(2x^4-4x^3)-(8x^2-16x)-(60x+120)=0}}}

{{{x^4(x-2)-11x^2(x-2)-2x^3(x-2)-8x(x-2)-60(x+2)=0}}}

{{{(x-2)(x^4-2x^3-11x^2-8x-60)=0}}}

{{{(x-2)(x^4-2x^3-15x^2+4x^2-8x-60)=0}}}

{{{(x-2)((x^4+4x^2)-(2x^3+8x)-(15x^2+60))=0}}}

{{{(x-2)(x^2(x^2+4)-2x(x^2+4)-15(x^2+4))=0}}}

{{{(x-2)(x^2 + 4)(x^2 - 2 x - 15)=0}}}

{{{(x-2)(x^2 + 4)(x^2 - 5x+3x - 15)=0}}}

{{{(x-2)(x^2 + 4)((x^2 - 5x)+(3x - 15))=0}}}

{{{(x-2)(x^2 + 4)(x(x - 5)+3(x - 5))=0}}}

{{{(x - 2) (x + 3) (x - 5) (x^2 + 4)=0}}}


the coaster returns to ground level at {{{x = 2}}}, {{{x=5}}}, {{{x = -3}}}, the three real zeros of the function



do all the roots of the polynomial {{{x^5-4x^4-7x^3+14x^2-44x+120}}} make sense in regards to a roller coaster engineering project problem????


the roots of the polynomial mathematically make sense in regards to a roller coaster engineering project problem because these values ({{{x = 2}}}, {{{x=5}}}, {{{x = -3}}}) mean the ride starts on the ground


but, mathematical end behavior of this function {{{does}}} {{{not}}} {{{make}}} {{{sense }}}{{{ practically}}}



function has local maximum and minimum where first derivative is equal to zero:

and it is  {{{5 x^4 - 16 x^3 - 21 x^2 + 28 x - 44=0}}} at

{{{x= -1.89}}}
{{{x=4.03}}}

so, function is
{{{highlight(Increasing)}}} on:  ({{{-infinity}}},{{{-1.89}}}),({{{4.03}}},{{{infinity}}})

{{{highlight(Decreasing)}}} on: ({{{-1.89}}},{{{4.03}}})


{{{ graph( 600, 600, -10, 10, -250, 250,(x - 2) (x + 3) (x - 5) (x^2 + 4)) }}}